Second order stabilized two-step Runge–Kutta methods

Abstract

Stabilized methods for the numerical solution of ODEs, also called Chebyshev methods, are explicit methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. In this paper we present explicit two-step Runge–Kutta–Chebyshev methods of order two, which have more than 2.3 times larger stability intervals than the analogous one-step methods. Explicit formulae for stability intervals are derived, as well as an effective recurrent scheme for calculation of methods’ coefficients for arbitrary number of stages. Our numerical experiments confirm the accuracy and stability properties of the proposed methods and show that at least in the case of constant time steps they can compete with the well-known ROCK2 method.